https://arxiv.org/pdf/2302.09519
It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be±22–√ instead of±2 , thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with.
Dr. Joy Christian obtained his Ph.D. from Boston University in Foundations of Quantum Theory in 1991 under the supervision of the renowned philosopher and physicist Professor Abner Shimony. He then received a Junior Research Fellowship from the Wolfson College of the University of Oxford, where he remained affiliated both with the college and a number of departments of the university until 2014. Currently he is the Scientific Director of the Einstein Centre for Local-Realistic Physics in Oxford. He has also been an invited member of the prestigious Foundational Questions Institute (FQXi), and a Long Term Visiting Professor at the Perimeter Institute for Theoretical Physics, Canada.
https://einstein-physics.org/about-the-centre/
the eigenvalues of non-commuting observables such as σx and σy do not add linearly, as we noted above. Consequently, the additivity relation (11) that holds for quantum states would not hold for the dispersion-free states
It assumes, without proof, that linear additivity of integrals leading to (23) can be meaningfully applied, not only to the eigenvalues of commuting observables but also to the eigenvalues of non-commuting observables that cannot be measured simultaneously. But, as we saw in the paragraph following equation (13), this assumption is quite mistaken.
However, as we noted around (13) and will be further demonstrated in Section VII, the built-in linear additivity of integrals is physically meaningful only for simultaneously measurable or commuting observables [31, 33]. It is, therefore, not legitimate to invoke it at step (23) without proof.
In fact, for non-commuting observables that are not simultaneously measurable,
justification of (23) or (32) by appealing to the built-in linear additivity of integrals leads to incorrect equality between unequal physical quantities.
A necessary step that would prove the consistency of the built-in linear additivity of anti-derivatives with the non-additivity of expectation values for the non-commuting observables. In equation (40) of Section VII below we will see the difference between the eigenvalue of the summed operator and the sum of individual eigenvalues explicitly. It will demonstrate how, in hidden variable theories equation (23) or (32) involving averages of eigenvalues ends up equating unequal averages of physical quantities in general.
It has to be a Two-Sphere - NOT a Zero-Sphere (as Bell wrongly assumed)....
Thus, what is brought out in it is the oversight of the non-commutative or Clifford-algebraic attributes of the physical space in Bell’s derivation of the bounds ±2 in (26). Non-locality or non-reality is necessitated only if one erroneously insists on linear additivity (23) of eigenvalues of non-commuting observables for each individual dispersion-free state | Ψ, λ).
Although the Product Rule (50) goes back to the pioneering investigations by von Neumann [3] and is universally accepted, since it plays a crucial role in the proof of GHZ’s argument, let me demonstrate that it does not hold for non-commuting operators.
what is asserted by (52) is not that ω1(λ), ω2(λ), and ˜ω(λ) cannot exist simultaneously as required by realism, but rather that the relationship among the three cannot be multiplicative if the operators Ω1 and Ω2 do not commute.
For observables that are not simultaneously measurable or commuting, the built-in linear additivity of integrals in step (23) or (32) leads to incorrect equality between averages of unequal physical quantities. Therefore, the view that this step is a harmless mathematical step is mistaken. It is, in fact, an unjustified assumption that is equivalent to the very thesis of the theorem to be proven, and is valid only in classical physics and/or for commuting observables.
The authors implicitly assume a multiplicative expectation function as specified by the Product Rule (50), even for the non-commuting observables (54) involved in their thought experiment. They thereby assume their conclusion in a different guise in the premisses of their argument.
Bell’s theorem is useful, however, for ruling out classical local theories. By relying on the assumption (23), which is valid for classical theories [31], it proves that no classical local theory can reproduce all of the predictions of quantum mechanics. But no serious hidden variable theories I am aware of have ever advocated returning to classical physics [26].
In this paper, I have focused on a formal or logical critique of Bell’s theorem. Elsewhere [10, 14, 16], I have developed a comprehensive local-realistic framework for understanding quantum correlations in terms of the geometry of the spatial part of one of the well-known solutions of Einstein’s field equations of general relativity — namely, that of a quaternionic 3-sphere — taken as a physical space within which we are confined to perform Bell-test experiments. This framework is based on Clifford algebra and thus explicitly takes the non-commutativity of observables into account. It thus shows, constructively, that contextually local hidden variable theories are not ruled out by Bell-test experiments.
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